Modelling Conformational Flexibility in a Spectrally Addressable Molecular Multi‐Qubit Model System

Abstract Dipolar coupled multi‐spin systems have the potential to be used as molecular qubits. Herein we report the synthesis of a molecular multi‐qubit model system with three individually addressable, weakly interacting, spin 1/2 centres of differing g‐values. We use pulsed Electron Paramagnetic Resonance (EPR) techniques to characterise and separately address the individual electron spin qubits; CuII, Cr7Ni ring and a nitroxide, to determine the strength of the inter‐qubit dipolar interaction. Orientation selective Relaxation‐Induced Dipolar Modulation Enhancement (os‐RIDME) detecting across the CuII spectrum revealed a strongly correlated CuII‐Cr7Ni ring relationship; detecting on the nitroxide resonance measured both the nitroxide and CuII or nitroxide and Cr7Ni ring correlations, with switchability of the interaction based on differing relaxation dynamics, indicating a handle for implementing EPR‐based quantum information processing (QIP) algorithms.


4-(10,15,20-triphenylporphyrin-5-yl)aniline -H2TPPNH2 (2)
SnCl2.2H2O (6.97 g, 36.67 mmol) was added to a green suspension of crude H2TPPNO2 (~1 g) in conc. HCl (130 mL) at room temperature. The mixture was heated, with stirring, to 65°C for 2 h. The reaction mixture was then allowed to cool, slowly added to ice-cold deionised H2O, and the pH adjusted to 9 using a saturated solution of NH4OH. The aqueous phase was then extracted with DCM (3 x 100 mL) and the organic phases combined and dried on anhydrous Na2SO4,

S.2 -Crystallographic data
Single crystal X-Ray diffraction data for [2]-rotaxane-TEMPO (8) was collected at a temperature of 100 K using a dual wavelength Rigaku FR-X with Cu-Kα radiation equipped with a HypixHE6000 detector and an Oxford Cryosystems nitrogen flow gas system. Data was measured using GDA and CrysAlisPro suite of programs. Single crystal XRD structures were visualised in Mercury software. The full details for this structure is available in pre-print. 12 Attempts to crystallise (9) were unsuccessful in a variety of solvent mixtures and by various crystallisation methods.

Crystal structure determinations and refinements
Single crystal X-Ray diffraction (XRD) data were processed and reduced using CrysAlisPro suite of programmes.
Absorption correction was performed using empirical methods (SCALE3 ABSPACK) based upon symmetry-equivalent reflections combined with measurements at different azimuthal angles. 4 The crystal structure was solved and refined against all F 2 values using the SHELXL and Olex 2 suite of programmes. 5 All atoms in crystal structures were refined anisotropically with the exception of the hydrogen atoms that were placed in the calculated idealized positions for all crystal structures. The pivalate and dimethylacrylate ligands, and threads in crystal structures were disordered and modelled over two positions, using structural same distance (SADI) and distance fix (DFIX) Shelxl restraints commands.
The atomic displacement parameters (adp) of the ligands have been restrained using similar Ueq and rigid bond (SIMU) and Similar Ueq (SIMU) restraints.

S.3 -EPR measurements
Samples of CuTPP-{Cr7Ni}-TEMPO (9) for pulsed EPR were prepared to a final concentration of ca. 200 µM in a 1:1:1 anhydrous mixture of d8-toluene:d8-THF:CDCl3 in 3 mm and 5 mm (o.d.) quartz tubes. Continuous Wave (CW) samples were prepared to a concentration of 1 mM in a 1:1:1 mixture of toluene:THF:CHCl3. Anhydrous deuterated solvents were obtained from Sigma-Aldrich as sealed ampoules and used as purchased, or purged with nitrogen (BOC) and passed through columns containing activated alumina and molecular sieves. The samples were degassed by three freeze-pumpthaw cycles, flame sealed, flash-frozen and then stored in liquid nitrogen. Pulsed EPR data were collected at Q-band on a Bruker ELEXSYS 580 FT spectrometer using a 3 mm Dual Mode Resonator (T2), and at X-band on a Bruker ELEXSYS E580X FT spectrometer using a 5 mm Split-Ring Resonator (MS5). Low temperature measurements were collected using a Cryogenic cryogen free variable temperature cryostat, or an Oxford Instruments cryostat incorporating a closed helium circuit. All pulses were amplified via a pulsed travelling wave tube (TWT) amplifier. Spectral simulations and analysis of spectra were performed in MATLAB R2020a, using Easyspin 5.2.30 simulation software [6] and were phase and background corrected to obtain Form factors before Fourier transformation to obtain a Pake pattern. For orientation independent data analysis, Tikhonov Regularization using the DeerAnalysis2019 routine was used to obtain the corresponding distance distributions. Orientation dependent simulations were carried out using a modified version of the algorithm reported by Lovett et al. [7] (Full details and GitHub repository link in S.5). , determined by simulation of the Q-band (34 GHz) CW EPR spectrum at 5 K using the Easyspin routine 'pepper' implented in MATLAB. The simulation parameters below were used to input spin system data for the orientation dependent simulations detailed in S.5. The spin Hamiltonian parameters obtained from fitting the simulation to the experimental data are in line with those reported for the corresponding component spin centres; Cr7Ni ring, [8] nitroxide, [9] Cu(II) porphyrin. [10] Simulation Parameter  Spin-spin relaxation measurements (T2) of (9) at Q-band (34 GHz) -3 K

Nitroxide -Cr7Ni
Magnetic  (Left) Tau averaged background RIDME trace with Tmix = 10 μs (equal to less than half the T1 of the Cr7Ni ring, containing only nitrogen and deuterium ESEEM contributions) for Cuxy -Cr7Ni RIDME, with the detection field centred at 1183.5 mT. The 10 μs tau-averaged background trace was subsequently divided out from the Cuxy -Cr7Ni RIDME at Tmix =120 μs to give the ESEEM-suppressed RIDME trace in the main text. (Right) The non-tau averaged RIDME trace with Tmix = 10 μs. A background measurement with the same Tmix block of 10 μs was obtained for each field position measured.
The supplementary orientation selective RIDME data are presented as raw traces and in the experimental time and frequency domains after division of a background trace (Tmix = 10 μs), phasing and background correction, and for Fig   S.3.7, the distance domain achieved after Tikhonov regularizationas this is dependent on g-values the obtained distance distributions are purely to visualise the suggested distance distribution between spin centres, and further, may contain artefacts owing to orientation selectivity between the centres, even after averaging.

VT-RIDME Cu(II)-Cr7Ni (34 GHz)
Variable temperature RIDME detecting on the Cu(II) spin gave Cu(II)-Cr7Ni RIDME signals up to 4 K, after which the traces become dominated by Electron Spin Envelope Echo Modulation (ESEEM) contributions, even with a sacrificial background trace.  The three pulse DEER sequence presented in the main text was chosen to afford greater sensitivity due to reduced transverse relaxation, however suffers from a dead-time around T = 0 of ca. 50 ns due to a temporal overlap of pump and detection pulses. When the oscillation of the dipolar interaction is expected to be large this dead-time becomes negligible.

DEER experimental details at X-band
To verify this assumption, a dead-time free four pulse DEER sequence was measured under the same experimental conditions and no appreciable difference, beyond a more accurately defined modulation depth, was found between the spectra. Further, measurements pumping on the maximum of the nitroxide resonance showed little to no orientation selectivity. Three pulse orientation selective DEER (os-DEER) measurements were performed (X-band, 15 K and 50 K) to investigate the Cu(II)-nitroxide dipolar interaction. Trends in the orientation selectivity lead to a narrower distribution in the dipolar interactions as the frequency offset between the pump and detection pulses is increased (Fig. S.3.11., a-c), however bandwidth limitations preclude fully sampling the gz component of the Cu(II) spectrum and isolating it from other orientations. In all cases, the dipolar interactions are less well-defined than for the Cu(II)-Cr7Ni interactions, as previously reported on a Cu(II) porphyrinnitroxide model system at X-band. [11] The relative positions of simulated nitroxide spin density is plotted relative to the

Cu(II)-NO RIDME at 50 K (Q-band)
To complement the Cu(II)-nitroxide os-DEER data presented in both the main text (15 K) and the supplementary information (50 K), os-RIDME at 50 K was performed detecting across the nitroxide spectrum. At 50 K the Cr7Ni ring loses its well defined spin ½ ground state and as such is effectively EPR silentthus relaxation of the ring does not need to be taken into consideration upon analysis of high temperature RIDME. These results are consistent with the data obtained from the 3P-DEER experiment and further demonstrate minimal orientation selectivity across the nitroxide spectrum, suggesting that there is a higher level of flexibility between these two spins and hence a higher number of conformational distributions between the Cu(II) spin centre and the nitroxide spin, as seen in the orientational analysis of the os-DEER experiments.

S.4 -Construction of model structures for orientation dependent simulations
Modifications of the single crystal X-ray diffraction (XRD) structure of the [2]-rotaxane-TEMPO (8) were carried out in Chemcraft (V. 1.8) to build a structural model of the title three spin compound (9). DFT (Density Functional Theory) calculations were carried out in ORCA (V.4.2.1) [12] in vacuo to obtain optimised geometries of model fragments of the three spin system to determine both the overall confirmation structure of the compound and the orientation of the g-matrix of the individual spin centres with respect to the molecular frame, and relative electron spin density information, necessary for the simulation of orientation dependent dipolar spectra. The g-matrix orientations were aligned and plotted using Avogadro (V.1.2.0). DFT results suggested accessible energy minima at the freezing point of the solvent at ca. 120°, 170° and 20° for dihedral angles A (blue), B (orange) and C (purple) respectivelythe input model for the orientation simulation was allowed to vary around these angles in order to generate the cones depicted in S.5.1. The DFT optimised single crystal XRD structure was presented in the main text was guided by these results, particularly the accessible orientations of dihedral A (blue).

Orientation of the Cu(II) porphyrin g-matrix with respect to the molecular frame
The DFT geometry optimised structure of the Cu(II) porphyrin fragment of complex (9) was used to compute the EPR parameters necessary for orientation dependent simulations. The axial g-matrix of Cu(II) porphyrins has been well reported, [13] and this was confirmed by DFT calculations implemented in ORCA using the BP functional in combination with basis sets def2-SVP, and def2-TZVP on the Cu(II) atom, an RI approximation, and an empirical dispersion correction to the energies. The gz component of the axial g-matrix is aligned perpendicular to the plane of the porphyrin ring, while the gx=gy components lie in the plane of the porphyrin ring.

Orientation of the Cr7Ni ring g-matrix with respect to the molecular frame
In the three spin system, the g-matrix of the Cr7Ni ring is assumed to be oriented according to previously reported singlecrystal studies on isolated anionic Cr7Ni rings which have shown that the gz component of the approximately axial g-matrix lies perpendicular to the plane of the ring, while the gx = gy component lies in the plane of the ring. [14] Figure S.4.4: Orientation of the g-matrix in the molecular frame of the Cr7Ni ring with the xyz matrix coordinates corresponding, in order, to the RGB colour sequence. The cationic ammonium thread has been removed for clarity. In the right image, the g-matrix has been shifted along the z-axis away from the plane of the Cr7Ni ring for clarity.

Orientation of the nitroxide g-matrix with respect to the molecular frame
The nitroxide spin has a well-documented g-matrix orientation and DFT results on the isolated single crystal XRD structure of the cationic ammonium thread using the BP functional in combination with the def2-SVP basis set, an RI approximation, and an empirical dispersion correction to the energies, corroborate previously reported studies. [15] The gz component of          (Fig. 3, b). The nitroxide fragment, H atoms, and tert-butyl groups have been removed.       Total difference between experimental and simulated traces vs. fitting iteration, leading to the least-square fits shown for Cu(II)nitroxide RIDME at 6 K as presented in the main text (Fig. 5, a).

os-RIDME Cu(II)-Cr7Ni ring at 3 K (34 GHz)orientation dependent simulations
Temperature dependent os-RIDME (34 GHz)  Validation of DEERAnalysis orientation independent distance distributions after Tikhonov Regularisation for background corrected and phased temperature dependent os-RIDME traces (Fig S.5.14). Black = best fits; Grey = error bars; Blue = lower error estimate corresponding to the mean value of the probability minus two times its standard deviation; Red = upper error estimate corresponding to the mean value plus two times the standard deviation.

Modulation depth and Form factor analysis in temperature dependent/multi-spin RIDME
Increased dipolar modulation depths are often associated with the presence of multi-spin effects (MSE) which result in sum and difference combinations to the pairwise dipolar frequency due to the higher probability of multiple spin-flipping events. In the case where the pure NO-Cr7Ni dipolar interaction is measured a modulation depth of 30% is obtained at the optimum Tmix (5 times T1 of the fastest relaxing spin) of the experiment. An increase in modulation depths is seen as we begin to extend Tmix and the temperature of the experiment, up to a maximum of ~50%. Due to the selectivity of the detection pulses and the large disparity in Tmix from that for the optimised trace (yellow), the enhanced modulation depths may be a result of the extended Tmix in which the fast relaxing spin of the targeted pairwise interaction has more time to undergo stochastic spin flipping events which increase the modulation of the detection spin echo. However, studying the range of distance distributions predicted from the datasets (Figure S.5.15) shows that as the mixing time is extended at 5 K there is both an increase in the signal predicted at ca. 3 nm, corresponding to the NO-Cu(II) interaction, and a shift to lower distances of the peak centred at ca. 1.35 nm, corresponding to the NO-Cr7Ni interaction. In particular the broadening of the peak centred at ca. 1.35 nm may be indicative of the presence of so-called ghost distances, which are a result of multi-spin interactions. Further work on the influence of multi-spin interactions in orthogonal spin models is planned. However, as the relative intensity of the two interactions seen in the distance distributions is always heavily skewed towards the presence of only one interaction, and the data measured at 3 K and 6 K are dominated by the NO-Cr7Ni and NO-Cu(II) interactions respectively, it is a reasonable first approximation to simply scale and combine the best fitting orientation dependent simulations of the pure pairwise interactions at 3 K and 6 K in order to fit the data at 5 K and intermediary Tmix as presented in the main text based off the % of the T1 value of the Cu(II) at 5 K that can be accessed (Fig. 5, amain text). Given a fitted T1 of 5.88 ms (Figure S.3.7) these values correspond to 17% of T1 (best fit scaled to 10% of Cu(II) 6 K RIDME fit), 255% of T1 (best fit scaled to 25% of Cu(II) 6 K RIDME fit), and 510% of T1 (best fit scaled to 50% of Cu(II) 6 K RIDME fit). The values used in the scaling of variable temperature RIDME detecting on the nitroxide spin provide reasonable fits to the data. If multi-spin effects were assumed, the simulation of orientation dependent dipolar taking into account the effect of more than one spin on the detection spin echo would be required in order to accurately fit the data.

S.6 -EPR-based Quantum Simulation of Decoherence
In order to perform a quantum simulation of the incoherent evolution of a quantum system, we consider an extended register containing additional qubits used to mimic the interaction of the system with the environment. In particular, the Cu(II) and nitroxide spins encode the "logical" qubits (L), while the Cr7Ni ring is the ancilla (A), representing the environment.
The incoherent dynamics of L is then simulated by first performing a coherent quantum simulation on the enlarged L+A system, in which L and A are entangled, followed by fast relaxation of A. This last step is equivalent to tracing out the environment degrees of freedom, thus making the simulated dynamics of L incoherent.
Here we focus, in particular, on the simulation of pure dephasing acting on a pair of system qubits. The steps involved in the procedure are the following: 1. Preparation of the initial state of L (see below). 2. Rotation of A by a pulse resonant with a Cr7Ni ring transition. The rotation angle controls the values of / 2 that we aim to simulate. This can be done with fast pulses, because of the need to rotate the Cr7Ni ring independently from the state of the other two. 3. Generate an entangling gate between A and one (controlled-NOT) or both (controlled-controlled-NOT) L qubits.
This requires long pulses to resolve the dipole-dipole interactions in order to make the Cr7Ni excitation dependent on the state of one or both L qubits. 4. Wait for the longitudinal relaxation of the Cr7Ni ring.
We now illustrate the preparation of the initial L state and the quantum simulation of decoherence in more detail.

A. Preparation of the initial state of L
A generic (factorised) state of the two "logical" qubits can be prepared by microwave pulses resonant with Cu(II) and/or nitroxide transitions.
In order to prepare an initial entangled state of the Cu(II) and nitroxide spin qubits (L), two-qubit gates among them are needed. The latter can be implemented by temporary excitations of the Cr7Ni spin. [16] More specifically, as long as the Cr7Ni ring is in the ground state the Cr7Ni ring-Cu(II) and Cr7Ni ring-nitroxide interactions only renormalise the external magnetic field, thus enabling single-qubit rotations of the two "logical qubits". Conversely, the excitation energy of Cr7Ni depends on the state of the other two spin qubits and therefore it is possible to perform excitations of the ring conditioned to the states of Cu(II) and nitroxide qubit pairs and hence two-qubit gates.
In particular, we can induce a full Rabi on the Cr7Ni ring only when both system qubits are ↓. In this way, a phase -1 is added only to the | ↓↓⟩ component of the two system qubit wavefunction, thus implementing a controlled-Z gate on L. By complementing this controlled-Z gate with rotations on L qubits, an entangled Bell state can be generated. The shorter T1 of the ring is not a major limitation here, because it is excited only during these two-qubit gates.

B. Quantum simulation of pure dephasing on L
On a single qubit coupled to an ancilla to mimic the environment, we can simulate the effect of pure dephasing on a generic system state |0⟩ + |1⟩ as follows: -Start with A qubit in |0⟩. Hence, the L+A state becomes ( |0⟩ + |1⟩) ⊗ |0⟩ -Rotate A by an angle , e.g.